Fourth-order Spatial Derivatives Research Articles

Nonlinear stability of shock-fronted travelling waves in reaction-nonlinear diffusion equations

Reaction-nonlinear diffusion PDEs can be derived as continuum limits of stochastic models for biological and ecological invasion. We numerically investigate the nonlinear stability of shock-fronted travelling waves arising in these RND PDEs, in the presence of a fourth-order spatial derivative multiplied by a small parameter ɛ that models high-order regularization. Once we have verified sectoriality of our linear operator, our task is reduced to checking spectral stability of our family of travelling waves. Motivated by the authors’ recent stability analysis of shock-fronted travelling waves under viscous relaxation, our numerical analysis suggests that near the singular limit, the associated eigenvalue problem for the linearized operator admits a fast–slow decomposition similar to that constructed by Alexander, Gardner, and Jones in the early 90s. In particular, our numerical results suggest a reduction of the complex four-dimensional eigenvalue problem into a real one-dimensional problem defined along the slow manifolds; i.e. slow eigenvalues defined near the tails of the shock-fronted wave for ɛ=0 govern the point spectrum of the linearized operator when 0<ɛ≪1.

An Improved Lax-Wendroff Scheme for Two-Dimensional Transient Thermal Simulation

The stability and accuracy of explicit high-order finite difference (HOFD) algorithms have been research hotspots in different fields. To improve the stability and accuracy of the HOFD algorithms in thermal simulations, we present a Lax-Wendroff high-order finite difference (LHOFD) algorithm to solve the 2D transient heat transfer equation in this paper and develop an improved LHOFD (IHOFD) algorithm to improve the stability of the LHOFD algorithm. The formulas of the general high-order central FD (HOCFD) coefficients and the truncation error coefficient as well as the high-order non-central FD (HONFD) coefficients and the truncation error coefficient of the fourth-order spatial derivative are derived concisely in a different way. Furthermore, a unified analytical formula of the general HOCFD and HONFD coefficients, which can calculate the spatial derivative of any integer order, is derived. A new strategy of combination with the HOCFD and HONFD approximations under the same high-order accuracy as the internal computational domain is proposed to calculate the mixed derivatives of the boundary domains with high accuracy, no additional computational cost, and easy implementation. Then, the accuracy analysis, stability analysis, and comparative analysis of numerical simulation results obtained by the LHOFD and IHOFD algorithms with the exact solution show the correctness and validity of the proposed algorithms and their stability formulas, and the advantages of the proposed algorithms. The proposed algorithms are valid under both symmetric and asymmetric boundary conditions. The stability factor of the LHOFD algorithm is slightly higher than that of the conventional algorithm. The stability factor of the IHOFD algorithm is twice that of the conventional algorithm, and the maximum absolute error of the thermal simulation is within 0.015 (°C).

A highly efficient semi-implicit corrective SPH scheme for 2D/3D tumor growth model

In this work, a semi-implicit corrective SPH method is proposed to solve the multi-component Cahn–Hilliard equation with fourth-order derivatives, and it is further used to predict the high-dimensional tumor growth model. The scheme is motivated by: (a) the fourth-order spatial derivative is discretized continuously by a corrective SPH formula for approximating second-order derivative twice, and the Neumann boundary is imposed by a ghost technique; (b) the temporal direction is approximated by the implicit scheme, and an iterative concept is employed to handle the above implicit form; (c) the multi-CPUs MPI parallelization is adopted to reduce the computing cost. Firstly, the second-order convergence rate of the proposed method for 2D/3D equation is shown and discussed by two analytical examples, and the mass conservation and energy properties are also demonstrated. Secondly, the efficiency of the proposed approach for multi-phase separation phenomenon is illustrated in an irregular domain. Finally, the 2D/3D tumor growth evolution at a short time is predicted by the proposed method and qualitatively compared with other numerical results. The numerical experiments show that the proposed scheme for phase-separation phenomenon or tumor growth model is highly efficient and reliable.

Peridynamics with strain gradient for modeling carbon nanotube under static and dynamic loading

This study couples peridynamic (PD) theory with strain gradient elasticity (SGE) theory to investigate the combined effect of PD and SGE length scale parameters on size effect. The SGE theory introduces a length scale parameter in the stress–strain relations to account for size effect. The solution to the classical form of SGE equation of motion requires two additional non-classical boundary conditions arising from the presence of fourth-order spatial derivatives. The wave dispersions level off as the wave number increases as observed in real materials. The PD theory allows for nonlocal interactions of a material point within its horizon which serves as the PD length scale parameter. The equation of motion emerges in the form of an integral equation and the internal forces are expressed through nonlocal interactions (bonds) between the material points within a continuous body. It is a reformulation of classical continuum mechanics which is free of spatial derivatives. The numerical results concern a previously considered carbon nanotube (CNT). Under quasi-static axial loading, the results indicate an increased strengthening effect along the length of the tube. Under dynamic loading, its longitudinal vibration response is captured through explicit time integration. The numerical predictions capture the reference solutions corresponding to the classical SGE equation.

An efficient pure meshless method for phase separation dominated by time-fractional Cahn–Hilliard equations

In this paper, an efficient Twice Finite Point-set Method (TFPM) coupled with difference scheme is proposed to solve the time-fractional Cahn–Hilliard (TF-CH) equation, and then it is extended to predict the phase separation process under nonlocal memory dominated by two-component TF-CH equations for the first time. The proposed meshless schemes are motivated by the following: (a) a high-order accuracy difference scheme is employed to approximate the time Caputo fractional derivative; (b) the fourth-order spatial derivative is divided into two second-order derivatives, and it is discretized by the FPM scheme continuously twice based on Taylor expansion and weighted least squares; (c) the Neumann boundary can be accurately imposed on the FPM scheme. In the numerical experiments, the error and numerical convergence of the proposed meshless method are first tested, which has near second-order convergent rate. Subsequently, the evolution of phase separation under memory dominated by single CH equation versus time is numerically investigated by the proposed method, and compared with the results in other literatures. The influence of fractional parameter on the separation phenomena is also discussed. Finally, the proposed method is used to predict the phase separation process under different parameters dominated by coupled TF-CH equations. All the numerical results show that the proposed coupled method is accurate in solving the TF-CH and efficient in predicting the phase separation evolution.

Modulation instability analysis and optical solutions of an extended (2+1)-dimensional perturbed nonlinear Schrödinger equation

We examine an extended (2+1)-dimensional perturbed nonlinear Schrödinger equation with Kerr law nonlinearity in a nano-optical fiber with fourth-order spatial derivatives in this paper. We develop various optical solutions to the equation under study using the Jacobi elliptic expansion method. We investigate the consequences of nonlinearity and spatial dispersion in spatial directions x and y. Furthermore, by investigating the stability condition of this nonlinear equation, we derive the linear stability analysis. The innovative evolutionary behaviors in the 3-dimensional profile are provided under constraint circumstances to indicate the wave propagation direction.

Compact expression of dyadic Green's function for homogeneous dielectric half‐space and its efficient double integral representation

The dyadic Green's function can be represented via spatial derivatives of two Sommerfeld integrals when an infinitesimal electric dipole and an observation point are located above an infinite planar dielectric interface. The two dielectric spaces are assumed to be homogeneous. Recently, new representations have been proposed for two Sommerfeld integrals and their complete uniform asymptotic expansions. The new formulation for the dyadic Green's function contains fourth-order spatial derivatives, so its explicit expression is lengthy and not compact. Thus, a compact form of the dyadic Green's function was formulated by converting the fourth-order spatial derivatives into second-order ones. Based on the new representation, a numerically efficient double integral expression of Green's function was obtained and numerically verified. In addition, five components among nine total electric field components were analytically evaluated in terms of the incomplete Hankel function when the source and observation point both are on the interface.

Optimal decay rate for higher–order derivatives of solution to the 3D compressible quantum magnetohydrodynamic model

Abstract We investigate optimal decay rates for higher–order spatial derivatives of strong solutions to the 3D Cauchy problem of the compressible viscous quantum magnetohydrodynamic model in the H 5 × H 4 × H 4 framework, and the main novelty of this work is three–fold: First, we show that fourth order spatial derivative of the solution converges to zero at the L 2 - rate ( 1 + t ) - 11 4 {L^2} - {\rm{rate}}\,{(1 + t)^{- {{11} \over 4}}} , which is same as one of the heat equation, and particularly faster than the L 2 - rate ( 1 + t ) - 5 4 {L^2} - {\rm{rate}}\,{(1 + t)^{- {5 \over 4}}} in Pu–Xu [Z. Angew. Math. Phys., 68:1, 2017] and the L 2 - rate ( 1 + t ) - 9 4 {L^2} - {\rm{rate}}\,{(1 + t)^{- {9 \over 4}}} , in Xi–Pu–Guo [Z. Angew. Math. Phys., 70:1, 2019]. Second, we prove that fifth–order spatial derivative of density ρ converges to zero at the L 2 - rate ( 1 + t ) - 13 4 {L^2} - {\rm{rate}}\,{(1 + t)^{- {{13} \over 4}}} , which is same as that of the heat equation, and particularly faster than ones of Pu–Xu [Z. Angew. Math. Phys., 68:1, 2017] and Xi–Pu–Guo [Z. Angew. Math. Phys., 70:1, 2019]. Third, we show that the high-frequency part of the fourth order spatial derivatives of the velocity u and magnetic B converge to zero at the L 2 - rate ( 1 + t ) - 13 4 {L^2} - {\rm{rate}}\,{(1 + t)^{- {{13} \over 4}}} , which are faster than ones of themselves, and totally new as compared to Pu–Xu [Z. Angew. Math. Phys., 68:1, 2017] and Xi–Pu–Guo [Z. Angew. Math. Phys., 70:1, 2019].

Heterogeneous Diffusion, Stability Analysis, and Solution Profiles for a MHD Darcy–Forchheimer Model

In the presented analysis, a heterogeneous diffusion is introduced to a magnetohydrodynamics (MHD) Darcy–Forchheimer flow, leading to an extended Darcy–Forchheimer model. The introduction of a generalized diffusion was proposed by Cohen and Murray to study the energy gradients in spatial structures. In addition, Peletier and Troy, on one side, and Rottschäfer and Doelman, on the other side, have introduced a general diffusion (of a fourth-order spatial derivative) to study the oscillatory patterns close the critical points induced by the reaction term. In the presented study, analytical conceptions to a proposed problem with heterogeneous diffusions are introduced. First, the existence and uniqueness of solutions are provided. Afterwards, a stability study is presented aiming to characterize the asymptotic convergent condition for oscillatory patterns. Dedicated solution profiles are explored, making use of a Hamilton–Jacobi type of equation. The existence of oscillatory patterns may induce solutions to be negative, close to the null equilibrium; hence, a precise inner region of positive solutions is obtained.

Bright and dark optical solitons of the (2+1)-dimensional perturbed nonlinear Schrödinger equation in nonlinear optical fibers

In this work, we study an extended (2+1)-dimensional perturbed nonlinear Schrödinger equation (P-NLS) with Kerr law nonlinearity in a nano optical fiber. The extended model includes fourth-order spatial derivatives. We study the influence of the nonlinearity and spatial dispersions effects given in spatial directions x and y. Various types of optical soliton solutions, that describe different structures of optical solutions, such as bright solitons, dark solitons, exponential, rational, and others as well, are extracted. The study is carried out by means of useful soliton ansatzes to retrieve this variety of optical soliton solutions. The obtained findings demonstrate that the implemented techniques are capable of identifying the exact solutions of other nonlinear evolution equations that arise in a range of scientific and engineering fields.

A Smoothing Scheme for Seismic Wave Propagation Simulation with a Mixed FEM of Diffusionized Wave Equation

In this paper, we propose a smoothing scheme for seismic wave propagation simulation. The proposed scheme is based on a diffusionized wave equation with the fourth-order spatial derivative term. So, the solution requires higher regularity in the usual weak formulation. Reducing the diffusionized wave equation to a coupled system of diffusion equations yields a mixed FEM to ease the regularity. We mathematically explain how our scheme works for smoothing. We construct a semi-implicit time integration scheme and apply it to the wave equation. This numerical experiment reveals that our scheme is effective for filtering short wavelength components in seismic wave propagation simulation.

A variety of bright and dark optical soliton solutions of an extended higher-order Sasa–Satsuma equation

In this work we examine an extended nonlinear (2+1)-dimensional Sasa–Satsuma equation that covers the effect of perturbations. The extended model includes fourth-order spatial derivatives. We study the influence of the nonlinearity and spatial dispersions effects given in spatial directions x and y. We formally derive a variety of optical soliton solutions, that include bright, dark, exponential, singular, and rational solutions. We employ useful techniques to retrieve this variety of optical soliton solutions.

Localized patterns in a generalized Swift–Hohenberg equation with a quartic marginal stability curve

Abstract In some pattern-forming systems, for some parameter values, patterns form with two wavelengths, while for other parameter values, there is only one wavelength. The transition between these can be organized by a codimension-three point at which the marginal stability curve has a quartic minimum. We develop a model equation to explore this situation, based on the Swift–Hohenberg equation; the model contains, amongst other things, snaking branches of patterns of one wavelength localized in a background of patterns of another wavelength. In the small-amplitude limit, the amplitude equation for the model is a generalized Ginzburg–Landau equation with fourth-order spatial derivatives, which can take the form of a complex Swift–Hohenberg equation with real coefficients. Localized solutions in this amplitude equation help interpret the localized patterns in the model. This work extends recent efforts to investigate snaking behaviour in pattern-forming systems where two different stable non-trivial patterns exist at the same parameter values.

Application of new quintic polynomial B-spline approximation for numerical investigation of Kuramoto\u2013Sivashinsky equation

A spline is a piecewise defined special function that is usually comprised of polynomials of a certain degree. These polynomials are supposed to generate a smooth curve by connecting at given data points. In this work, an application of fifth degree basis spline functions is presented for a numerical investigation of the Kuramoto–Sivashinsky equation. The finite forward difference formula is used for temporal integration, whereas the basis splines, together with a new approximation for fourth order spatial derivative, are brought into play for discretization in space direction. In order to corroborate the presented numerical algorithm, some test problems are considered and the computational results are compared with existing methods.

Fractional Model of Couple Stress Fluid for Generalized Couette Flow: A Comparative Analysis of Atangana–Baleanu and Caputo–Fabrizio Fractional Derivatives

Different from a Newtonian fluid, couple stress uid (CSF) includes a new material constant, which is responsible for couple stress and the lubricant viscosity. This material constant comes with the fourth-order spatial derivative term, and due to this higher-order derivative term in the momentum equation, this uid (CSF) is comparatively less investigated even for the classical uid problems. This paper aims to study the fractional model of CSF, based on the Atangana-Baleanu (AB) fractional derivatives de nition. Since this AB de nition is new, therefore, for the sake of comparison and correctness, this problem is also solved using the Caputo-Fabrizio (CF) fractional derivative de nition. The CSF is considered to flow between two parallel plates, one of which is at rest and the other is moving with constant velocity. The external pressure gradient is also applied. This type of ow situations is usually called generalized Couette ow. The problem is rst written in dimensionless form and then solved for the exact solution using the Laplace transform and the nite Fourier sine transform. The CSF velocity obtained via an AB fractional derivative is compared with the CSF velocity obtained via a CF fractional derivative approach, and the results obtained are shown graphically. The CSF results for interesting uid parameters are displayed in various graphs for both the AB and CF fractional derivatives. It is observed that the CSF velocities obtained with the AB and CF fractional derivatives are the same for unit time. For time less than one and greater than one, variation in CSF velocities is observed. In limiting sense, the present CSF solutions are reduced to a similar Newtonian uid problem solution in the absence of external pressure gradient.

Krylov Implicit Integration Factor Methods for Semilinear Fourth-Order Equations

Implicit integration factor (IIF) methods were developed for solving time-dependent stiff partial differential equations (PDEs) in literature. In [Jiang and Zhang, Journal of Computational Physics, 253 (2013) 368–388], IIF methods are designed to efficiently solve stiff nonlinear advection–diffusion–reaction (ADR) equations. The methods can be designed for an arbitrary order of accuracy. The stiffness of the system is resolved well, and large-time-step-size computations are achieved. To efficiently calculate large matrix exponentials, a Krylov subspace approximation is directly applied to the IIF methods. In this paper, we develop Krylov IIF methods for solving semilinear fourth-order PDEs. As a result of the stiff fourth-order spatial derivative operators, the fourth-order PDEs have much stricter constraints in time-step sizes than the second-order ADR equations. We analyze the truncation errors of the fully discretized schemes. Numerical examples of both scalar equations and systems in one and higher spatial dimensions are shown to demonstrate the accuracy, efficiency and stability of the methods. Large time-step sizes that are of the same order as the spatial grid sizes have been achieved in the simulations of the fourth-order PDEs.

Method of Lines Transpose: Energy Gradient Flows Using Direct Operator Inversion for Phase-Field Models

In this work, we develop an $\mathcal{O}(N)$ implicit real space method in 1D and 2D for the Cahn--Hilliard (CH) and vector Cahn--Hilliard (VCH) equations, based on the method of lines transpose (MOL$^{T}$) formulation. This formulation results in a semidiscrete time stepping algorithm, which we prove is gradient stable in the $H^{-1}$ norm. The spatial discretization follows from dimensional splitting and an $\mathcal{O}(N)$ matrix-free solver, which applies fast convolution to the modified Helmholtz equation. We propose a novel factorization technique, in which fourth-order spatial derivatives are incorporated into the solver. The splitting error is included in the nonlinear fixed point iteration, resulting in a high-order, logically Cartesian (line-by-line) update. Our method is fast but not restricted to periodic boundaries like the fast Fourier transform (FFT). The basic solver is implemented using the backward Euler formulation, and we extend this to both backward difference formula (BDF) stencils, singly diagonal implicit Runge--Kutta (SDIRK), and spectral deferred correction (SDC) frameworks to achieve high orders of temporal accuracy. We demonstrate with numerical results that the CH and VCH equations maintain gradient stability in one and two spatial dimensions. We also explore time-adaptivity, so that meta-stable states and ripening events can be simulated both quickly and efficiently.

Global well-posedness and decay of smooth solutions to the non-isothermal model for compressible nematic liquid crystals

The Cauchy problem for the three-dimensional non-isothermal model for compressible nematic liquid crystals is considered. Existence of global-in-time smooth solutions is established provided that the initial datum is close to a steady state (ρ¯,0,d¯,θ¯). By using the Lq–Lp estimates and the Fourier splitting method, if the initial perturbation is small in H3-norm and bounded in Lq (q∈[1,65)) norm, we obtain the optimal decay rates for the first and second order spatial derivatives of solutions. In addition, the third and fourth order spatial derivatives of director field d in L2-norm are achieved.

Solitary Wave Formation from a Generalized Rosenau Equation

A generalized viscous Rosenau equation containing linear and nonlinear advective terms and mixed third- and fifth-order derivatives is studied numerically by means of an implicit second-order accurate method in time that treats the first-, second-, and fourth-order spatial derivatives as unknown and discretizes them by means of three-point, fourth-order accurate, compact finite differences. It is shown that the effect of the viscosity is to decrease the amplitude, curve the wave trajectory, and increase the number and width of the waves that emerge from an initial Gaussian condition, whereas the linear convective term pushes the wave front towards the downstream boundary. It is also shown that the effect of the nonlinear convective term is to increase the steepness of the leading wave front and the number of sawtooth waves that are generated behind it, while that of the first dispersive term is to increase the number of waves that break up from the initial condition as the coefficient that characterizes this term is decreased. It is also shown that, for reasons of stability, the second dispersion coefficient must be much smaller than the first one and its effects on wave propagation are relatively small.

Necessary conditions for breathers on continuous media to approximate breathers on discrete lattices

The sine-Gordon (SG) partial differential equation (PDE) with an arbitrary perturbation is initially considered. Using the method of Kuzmak–Luke, we investigate the conditions, which the perturbation must satisfy, for a breather solution to be a valid leading-order asymptotic approximation to the perturbed problem. We analyse the cases of both stationary and moving breathers. As examples, we consider perturbing terms which include typical linear damping, periodic sinusoidal driving, and dispersion. The motivation for this study is that the mathematical modelling of physical systems often leads to the discrete SG system of ordinary differential equations, which are then approximated in the long wavelength limit by the continuous SG PDE. Such limits typically produce fourth-order spatial derivatives as correction terms. The new results show that the stationary breather solution is a consistent solution of both the quasi-continuum SG equation and the forced/damped SG system. However, the moving breather is only a consistent solution of the quasi-continuum SG equation and not the damped SG system.